Related papers: A note on the Erd\H{o}s distinct subset sums probl…

We find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$, for $p=\mathrm{const}$ and $p=n^{-\alpha+o(1)}$.

{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that…

We study families of subsets of $\omega$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that…

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this…

Let $A$ and $B$ be sets of words of length $n$ over some finite alphabet. Suppose that no suffix of a word in $A$ coincides with a prefix of a word in $B$. Then we show that the product of densities of $A$ and $B$ is upper bounded by…

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log…

Here we prove an asymptotically optimal lower bound on the information complexity of the k-party disjointness function with the unique intersection promise, an important special case of the well known disjointness problem, and the…

The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by…

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…

Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

Answering several questions of Duffus, Frankl and R\"odl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering…

A set $A$ is MSTD (more-sum-than-difference) if $|A+A|>|A-A|$. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{1,2,\ldots ,r\}$ as…

We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous…

We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovasz theta number and of…

We establish a majorization-based theory for bounding observables of waves with varied coherence. For any measurement, exact bounds are attained by the maximal and minimal elements in the set of input coherence spectra. The set's supremum…

We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$…

Let $B(2d-1, d)$ be the subgraph of the hypercube $\mathcal{Q}_{2d-1}$ induced by its two largest layers. Duffus, Frankl and R\"odl proposed the problem of finding the asymptotics for the logarithm of the number of maximal independent sets…